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Properties

Label 2-930-15.2-c1-0-15
Degree $2$
Conductor $930$
Sign $0.167 - 0.985i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (1.14 + 1.30i)3-s − 1.00i·4-s + (−1.36 + 1.77i)5-s + (1.72 + 0.112i)6-s + (−0.0800 − 0.0800i)7-s + (−0.707 − 0.707i)8-s + (−0.390 + 2.97i)9-s + (0.286 + 2.21i)10-s + 3.80i·11-s + (1.30 − 1.14i)12-s + (2.77 − 2.77i)13-s − 0.113·14-s + (−3.86 + 0.244i)15-s − 1.00·16-s + (−4.20 + 4.20i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.659 + 0.751i)3-s − 0.500i·4-s + (−0.610 + 0.791i)5-s + (0.705 + 0.0461i)6-s + (−0.0302 − 0.0302i)7-s + (−0.250 − 0.250i)8-s + (−0.130 + 0.991i)9-s + (0.0905 + 0.701i)10-s + 1.14i·11-s + (0.375 − 0.329i)12-s + (0.768 − 0.768i)13-s − 0.0302·14-s + (−0.998 + 0.0631i)15-s − 0.250·16-s + (−1.02 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.167 - 0.985i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.167 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53788 + 1.29817i\)
\(L(\frac12)\) \(\approx\) \(1.53788 + 1.29817i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.14 - 1.30i)T \)
5 \( 1 + (1.36 - 1.77i)T \)
31 \( 1 + T \)
good7 \( 1 + (0.0800 + 0.0800i)T + 7iT^{2} \)
11 \( 1 - 3.80iT - 11T^{2} \)
13 \( 1 + (-2.77 + 2.77i)T - 13iT^{2} \)
17 \( 1 + (4.20 - 4.20i)T - 17iT^{2} \)
19 \( 1 - 6.37iT - 19T^{2} \)
23 \( 1 + (4.35 + 4.35i)T + 23iT^{2} \)
29 \( 1 - 7.03T + 29T^{2} \)
37 \( 1 + (-3.86 - 3.86i)T + 37iT^{2} \)
41 \( 1 - 9.64iT - 41T^{2} \)
43 \( 1 + (-1.84 + 1.84i)T - 43iT^{2} \)
47 \( 1 + (-2.31 + 2.31i)T - 47iT^{2} \)
53 \( 1 + (-5.01 - 5.01i)T + 53iT^{2} \)
59 \( 1 - 8.84T + 59T^{2} \)
61 \( 1 + 7.40T + 61T^{2} \)
67 \( 1 + (-3.19 - 3.19i)T + 67iT^{2} \)
71 \( 1 + 0.739iT - 71T^{2} \)
73 \( 1 + (-6.88 + 6.88i)T - 73iT^{2} \)
79 \( 1 + 3.84iT - 79T^{2} \)
83 \( 1 + (-4.34 - 4.34i)T + 83iT^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + (4.93 + 4.93i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35331166881532802286687652991, −9.810546988873396948884082500626, −8.425461773930559065869861408813, −8.032516234533958886234631345668, −6.75627395867767525379156394261, −5.86074004490877930865527604083, −4.45130095990752005114975484351, −4.00766351961019388032974567564, −3.01585224086569052230868908087, −1.99519232090565666839004839677, 0.74408784152734889263708022674, 2.45159751492248250297860621618, 3.61340257846087872466537090137, 4.44415784443835345317567338977, 5.61698924534850371428898437944, 6.58115171799684293219506912680, 7.30050352130260791019293533502, 8.212852222423296426697827247765, 8.878922513250326977556811959533, 9.294980949072253785409511594244

Graph of the $Z$-function along the critical line